I need planes to coincide, as well as possessing marked freestyle line edges

Why do the planes, in this case, need to coincide? Couldn't you have created the model from a single contiguous mesh, and fold it using bones or deformation modifiers? This technique would help avoid planes coinciding in the same coordinate space and solve the artifact issues you're facing in this animation. Blender almost always rejects overlapping faces in rendering scenarios, and it usually considers it a bad topological practice. There are a few folding examples for different paper airplane designs online you can glean ideas from, but if you are going to keep this design strategy of individually animated objects, it's best to avoid overlapping faces.

The Blender Render used to have an option called Z-Offset material back in 2.5 days, which became active when Z-Trans was enabled. This might be your only hope in avoiding artifacts with overlapping geometry, but I have not used the Blender Render engine in years, and I never tested that option fully. You'd have to look it up to see if it would work for you, but the rule of thumb is generally to avoid overlapping faces at all costs, because they may never render predictably in Blender.

______________________________________

My Store
My Free Models
My Video Tutorials
My CG Animations
Instagram: @luxxeon3d
Facebook: https://www.facebook.com/luxxeon